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Theorem rpnnen1lem5 12019
Description: Lemma for rpnnen1 12021. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
rpnnen1lem.1 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
rpnnen1lem.2 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
rpnnen1lem.n ℕ ∈ V
rpnnen1lem.q ℚ ∈ V
Assertion
Ref Expression
rpnnen1lem5 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
Distinct variable groups:   𝑘,𝐹,𝑛,𝑥   𝑇,𝑛
Allowed substitution hints:   𝑇(𝑥,𝑘)

Proof of Theorem rpnnen1lem5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rpnnen1lem.1 . . . 4 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
2 rpnnen1lem.2 . . . 4 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
3 rpnnen1lem.n . . . 4 ℕ ∈ V
4 rpnnen1lem.q . . . 4 ℚ ∈ V
51, 2, 3, 4rpnnen1lem3 12017 . . 3 (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥)
61, 2, 3, 4rpnnen1lem1 12016 . . . . . 6 (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))
74, 3elmap 8089 . . . . . 6 ((𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ) ↔ (𝐹𝑥):ℕ⟶ℚ)
86, 7sylib 209 . . . . 5 (𝑥 ∈ ℝ → (𝐹𝑥):ℕ⟶ℚ)
9 frn 6229 . . . . . 6 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℚ)
10 qssre 11999 . . . . . 6 ℚ ⊆ ℝ
119, 10syl6ss 3773 . . . . 5 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ⊆ ℝ)
128, 11syl 17 . . . 4 (𝑥 ∈ ℝ → ran (𝐹𝑥) ⊆ ℝ)
13 1nn 11287 . . . . . . . 8 1 ∈ ℕ
1413ne0ii 4088 . . . . . . 7 ℕ ≠ ∅
15 fdm 6231 . . . . . . . 8 ((𝐹𝑥):ℕ⟶ℚ → dom (𝐹𝑥) = ℕ)
1615neeq1d 2996 . . . . . . 7 ((𝐹𝑥):ℕ⟶ℚ → (dom (𝐹𝑥) ≠ ∅ ↔ ℕ ≠ ∅))
1714, 16mpbiri 249 . . . . . 6 ((𝐹𝑥):ℕ⟶ℚ → dom (𝐹𝑥) ≠ ∅)
18 dm0rn0 5510 . . . . . . 7 (dom (𝐹𝑥) = ∅ ↔ ran (𝐹𝑥) = ∅)
1918necon3bii 2989 . . . . . 6 (dom (𝐹𝑥) ≠ ∅ ↔ ran (𝐹𝑥) ≠ ∅)
2017, 19sylib 209 . . . . 5 ((𝐹𝑥):ℕ⟶ℚ → ran (𝐹𝑥) ≠ ∅)
218, 20syl 17 . . . 4 (𝑥 ∈ ℝ → ran (𝐹𝑥) ≠ ∅)
22 breq2 4813 . . . . . . 7 (𝑦 = 𝑥 → (𝑛𝑦𝑛𝑥))
2322ralbidv 3133 . . . . . 6 (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
2423rspcev 3461 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
255, 24mpdan 678 . . . 4 (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦)
26 id 22 . . . 4 (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
27 suprleub 11243 . . . 4 (((ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦) ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
2812, 21, 25, 26, 27syl31anc 1492 . . 3 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥))
295, 28mpbird 248 . 2 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥)
301, 2, 3, 4rpnnen1lem4 12018 . . . . . . . . 9 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
31 resubcl 10599 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
3230, 31mpdan 678 . . . . . . . 8 (𝑥 ∈ ℝ → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
3332adantr 472 . . . . . . 7 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ)
34 posdif 10775 . . . . . . . . . 10 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
3530, 34mpancom 679 . . . . . . . . 9 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
3635biimpa 468 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < )))
3736gt0ne0d 10846 . . . . . . 7 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ≠ 0)
3833, 37rereccld 11106 . . . . . 6 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) ∈ ℝ)
39 arch 11535 . . . . . 6 ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) ∈ ℝ → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘)
4038, 39syl 17 . . . . 5 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘)
4140ex 401 . . . 4 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘))
421, 2rpnnen1lem2 12015 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)
4342zred 11729 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℝ)
44433adant3 1162 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) ∈ ℝ)
4544ltp1d 11208 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1))
4633, 36jca 507 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) → ((𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
47 nnre 11282 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
48 nngt0 11306 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 0 < 𝑘)
4947, 48jca 507 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
50 ltrec1 11164 . . . . . . . . . . . . 13 ((((𝑥 − sup(ran (𝐹𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
5146, 49, 50syl2an 589 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
5230ad2antrr 717 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
53 nnrecre 11314 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
5453adantl 473 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
55 simpll 783 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
5652, 54, 55ltaddsub2d 10882 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))))
5712adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ran (𝐹𝑥) ⊆ ℝ)
58 ffn 6223 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑥):ℕ⟶ℚ → (𝐹𝑥) Fn ℕ)
598, 58syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝐹𝑥) Fn ℕ)
60 fnfvelrn 6546 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑥) Fn ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6159, 60sylan 575 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥))
6257, 61sseldd 3762 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ∈ ℝ)
6330adantr 472 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
6453adantl 473 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
6512, 21, 253jca 1158 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦))
6665adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦))
67 suprub 11238 . . . . . . . . . . . . . . . . 17 (((ran (𝐹𝑥) ⊆ ℝ ∧ ran (𝐹𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑦) ∧ ((𝐹𝑥)‘𝑘) ∈ ran (𝐹𝑥)) → ((𝐹𝑥)‘𝑘) ≤ sup(ran (𝐹𝑥), ℝ, < ))
6866, 61, 67syl2anc 579 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) ≤ sup(ran (𝐹𝑥), ℝ, < ))
6962, 63, 64, 68leadd1dd 10895 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)))
7062, 64readdcld 10323 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ)
71 readdcl 10272 . . . . . . . . . . . . . . . . 17 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
7230, 53, 71syl2an 589 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
73 simpl 474 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
74 lelttr 10382 . . . . . . . . . . . . . . . . 17 (((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∧ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
7574expd 404 . . . . . . . . . . . . . . . 16 (((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
7670, 72, 73, 75syl3anc 1490 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
7769, 76mpd 15 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
7877adantlr 706 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
7956, 78sylbird 251 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝑥 − sup(ran (𝐹𝑥), ℝ, < )) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
8051, 79sylbid 231 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
8142peano2zd 11732 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℤ)
82 oveq1 6849 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → (𝑛 / 𝑘) = ((sup(𝑇, ℝ, < ) + 1) / 𝑘))
8382breq1d 4819 . . . . . . . . . . . . . . . . . 18 (𝑛 = (sup(𝑇, ℝ, < ) + 1) → ((𝑛 / 𝑘) < 𝑥 ↔ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
8483, 1elrab2 3523 . . . . . . . . . . . . . . . . 17 ((sup(𝑇, ℝ, < ) + 1) ∈ 𝑇 ↔ ((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
8584biimpri 219 . . . . . . . . . . . . . . . 16 (((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
8681, 85sylan 575 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
87 ssrab2 3847 . . . . . . . . . . . . . . . . . . . 20 {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ⊆ ℤ
881, 87eqsstri 3795 . . . . . . . . . . . . . . . . . . 19 𝑇 ⊆ ℤ
89 zssre 11631 . . . . . . . . . . . . . . . . . . 19 ℤ ⊆ ℝ
9088, 89sstri 3770 . . . . . . . . . . . . . . . . . 18 𝑇 ⊆ ℝ
9190a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ⊆ ℝ)
92 remulcl 10274 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
9392ancoms 450 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
9447, 93sylan2 586 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 · 𝑥) ∈ ℝ)
95 btwnz 11726 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 · 𝑥) ∈ ℝ → (∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥) ∧ ∃𝑛 ∈ ℤ (𝑘 · 𝑥) < 𝑛))
9695simpld 488 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 · 𝑥) ∈ ℝ → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
9794, 96syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
98 zre 11628 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℤ → 𝑛 ∈ ℝ)
9998adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ)
100 simpll 783 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
10149ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
102 ltdivmul 11152 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((𝑛 / 𝑘) < 𝑥𝑛 < (𝑘 · 𝑥)))
10399, 100, 101, 102syl3anc 1490 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥𝑛 < (𝑘 · 𝑥)))
104103rexbidva 3196 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥 ↔ ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥)))
10597, 104mpbird 248 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
106 rabn0 4122 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅ ↔ ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
107105, 106sylibr 225 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
1081neeq1i 3001 . . . . . . . . . . . . . . . . . 18 (𝑇 ≠ ∅ ↔ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
109107, 108sylibr 225 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ≠ ∅)
1101rabeq2i 3346 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝑇 ↔ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥))
11147ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑘 ∈ ℝ)
112111, 100, 92syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 · 𝑥) ∈ ℝ)
113 ltle 10380 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℝ ∧ (𝑘 · 𝑥) ∈ ℝ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
11499, 112, 113syl2anc 579 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
115103, 114sylbid 231 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥𝑛 ≤ (𝑘 · 𝑥)))
116115impr 446 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥)) → 𝑛 ≤ (𝑘 · 𝑥))
117110, 116sylan2b 587 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛𝑇) → 𝑛 ≤ (𝑘 · 𝑥))
118117ralrimiva 3113 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥))
119 breq2 4813 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑘 · 𝑥) → (𝑛𝑦𝑛 ≤ (𝑘 · 𝑥)))
120119ralbidv 3133 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑘 · 𝑥) → (∀𝑛𝑇 𝑛𝑦 ↔ ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥)))
121120rspcev 3461 . . . . . . . . . . . . . . . . . 18 (((𝑘 · 𝑥) ∈ ℝ ∧ ∀𝑛𝑇 𝑛 ≤ (𝑘 · 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦)
12294, 118, 121syl2anc 579 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦)
12391, 109, 1223jca 1158 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦))
124 suprub 11238 . . . . . . . . . . . . . . . 16 (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛𝑇 𝑛𝑦) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
125123, 124sylan 575 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
12686, 125syldan 585 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
127126ex 401 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < )))
12842zcnd 11730 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℂ)
129 1cnd 10288 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
130 nncn 11283 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
131 nnne0 11310 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
132130, 131jca 507 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
133132adantl 473 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
134 divdir 10964 . . . . . . . . . . . . . . . 16 ((sup(𝑇, ℝ, < ) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0)) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
135128, 129, 133, 134syl3anc 1490 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
1363mptex 6679 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V
1372fvmpt2 6480 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V) → (𝐹𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
138136, 137mpan2 682 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝐹𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
139138fveq1d 6377 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ → ((𝐹𝑥)‘𝑘) = ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘))
140 ovex 6874 . . . . . . . . . . . . . . . . . 18 (sup(𝑇, ℝ, < ) / 𝑘) ∈ V
141 eqid 2765 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))
142141fvmpt2 6480 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℕ ∧ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V) → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
143140, 142mpan2 682 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
144139, 143sylan9eq 2819 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹𝑥)‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
145144oveq1d 6857 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
146135, 145eqtr4d 2802 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = (((𝐹𝑥)‘𝑘) + (1 / 𝑘)))
147146breq1d 4819 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 ↔ (((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
14881zred 11729 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
149148, 43lenltd 10437 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ) ↔ ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
150127, 147, 1493imtr3d 284 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
151150adantlr 706 . . . . . . . . . . 11 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((((𝐹𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
15280, 151syld 47 . . . . . . . . . 10 (((𝑥 ∈ ℝ ∧ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
153152exp31 410 . . . . . . . . 9 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
154153com4l 92 . . . . . . . 8 (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → (𝑥 ∈ ℝ → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
155154com14 96 . . . . . . 7 (𝑥 ∈ ℝ → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
1561553imp 1137 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
15745, 156mt2d 133 . . . . 5 ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘) → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)
158157rexlimdv3a 3180 . . . 4 (𝑥 ∈ ℝ → (∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹𝑥), ℝ, < ))) < 𝑘 → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥))
15941, 158syld 47 . . 3 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) < 𝑥 → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥))
160159pm2.01d 181 . 2 (𝑥 ∈ ℝ → ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)
161 eqlelt 10379 . . 3 ((sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)))
16230, 161mpancom 679 . 2 (𝑥 ∈ ℝ → (sup(ran (𝐹𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹𝑥), ℝ, < ) < 𝑥)))
16329, 160, 162mpbir2and 704 1 (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  wss 3732  c0 4079   class class class wbr 4809  cmpt 4888  dom cdm 5277  ran crn 5278   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  supcsup 8553  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194   < clt 10328  cle 10329  cmin 10520   / cdiv 10938  cn 11274  cz 11624  cq 11989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-sup 8555  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-n0 11539  df-z 11625  df-q 11990
This theorem is referenced by:  rpnnen1lem6  12020
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